Define n-quantum vector space to be the algebra $$ {\mathbb C}_q^n := \mathbb{C}< x_i ~ | ~ i =1, \ldots, N>/<x_i x_j = q x_j x_i ~ | ~ i< j>. $$ For $q=1$, we get the usual polynomial ring in $n$-variables, and so, Serre's conjecture (Quillen–Suslin theorem) tells that every finitely generated projective module over ${\mathbb C}_1^n$ is free. How does this work for $q \neq 1$? Is there a $q$-deformed Quillen–Suslin theorem? The not a root of unity case is the most interesting to me.

Define n-quantum vector space to be the algebra $$ {\mathbb C}_q^n := \mathbb{C}\left< x_i \mid i =1, \ldots, N\right>/\left<x_i x_j = q x_j x_i \mid i<j\right>. $$ For $q=1$, we get the usual polynomial ring in $n$-variables, and so, Serre's conjecture (Quillen–Suslin theorem) tells that every finitely generated projective module over ${\mathbb C}_1^n$ is free. How does this work for $q \neq 1$? Is there a $q$-deformed Quillen–Suslin theorem? The not a root of unity case is the most interesting to me.

Define n-quantum vector space to be the algebra $$ {\mathbb C}_q^n := \mathbb{C}< x_i ~ | ~ i =1, \ldots, N>/<x_i x_j = q x_j x_i ~ | ~ i< j>. $$ For $q=1$, we get the usual polynomial ring in $n$-variables, and so, Serre's conjecture (Quillen–Suslin theorem) tells that every finitely generated projective module over ${\mathbb C}_1^n$ is free. How does this work for $q \neq 1$? Is there a $q$-deformed Quillen–Suslin theorem

Define n-quantum vector space to be the algebra $$ {\mathbb C}_q^n := \mathbb{C}< x_i ~ | ~ i =1, \ldots, N>/<x_i x_j = q x_j x_i ~ | ~ i< j>. $$ For $q=1$, we get the usual polynomial ring in $n$-variables, and so, Serre's conjecture (Quillen–Suslin theorem) tells that every finitely generated projective module over ${\mathbb C}_1^n$ is free. How does this work for $q \neq 1$? Is there a $q$-deformed Quillen–Suslin theorem? The not a root of unity case is the most interesting to me.